Legendre spectral element for plastic localization problems at large scale strains
https://doi.org/10.22405/2226-8383-2020-21-3-306-316
Abstract
In paper the method of spectral elements based on the Legendre polynomial for timeindependent
elastic-plastic plane problems at large strains is proposed. The method of spectral
elements is based on the variational principle (Galerkin’s method). The solution of these
problems has the phenomenon of localization of plastic deformations in narrow areas called
slip-line or shear band. The possibility of using a spectral element for the numerical solution of
these problems with discontinuous solutions is investigated. The yield condition of the material
is the von Mises criterion. The stresses are integrated by the radial return method by backward
implicit Euler scheme. The system of nonlinear algebraic equations is solved by the Newton’s
iterative method. A numerical solution is given of an example of stretching a strip weakened by
cuts with a circular base in a plane stress and plane deformed state. Kinematic fields and limit
load are obtained. Comparisons of numerical results with the analytical solution obtained for
incompressible media constructed by the method of characteristics are presented.
Keywords
spectral method,
localization phenomenon,
plasticity,
slip-line,
finite strains,
iterative Newton’s method
Supporting agencies: This work was supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of the basic part of the Government task for scientific activity (project No. 9.7446.2017 / 8.9) in the part related to the mathematical formulation of the problem, with financial support from the Russian Foundation of Basic Research and the Moscow Government (project No. 19-38-70001) in the part related to the development of the mathematical method and algorithm for problem solving, with financial support from the Russian Science Foundation (project No. 19-71-10008) in the part related to development and verification of software and analysis of calculation results.
About the Authors
Vladimir Anatolyevich Levin
Lomonosov Moscow State University
Russian Federation
Russian Federation
Doctor of physical and mathematical sciences, professor, Professor of the department of computational mechanics of the faculty of mechanics and mathematics
Konstasntin Moiseevich Zingerman
Tver State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor, head of chair of mathematical modeling and computational mathematics
Kirill Yurievich Krapivin
CAE Fidesys
Russian Federation
Russian Federation
Maksim Yakovlevich Yakovlev
Lomonosov Moscow State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, Associate Professor of the Department of Computational Mathematics of the Faculty of Mechanics and Mathematics
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Review
For citations:
Levin V.A., Zingerman K.M., Krapivin K.Yu., Yakovlev M.Ya. Legendre spectral element for plastic localization problems at large scale strains. Chebyshevskii Sbornik. 2020;21(3):306-316. (In Russ.) https://doi.org/10.22405/2226-8383-2020-21-3-306-316
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